The answer is different for each family. Start looking for another dog or cat if and when you feel you are ready. Don’t let other people control your timing… you need time to grieve, but you also need to allow yourself to move forward with life.
Remember, after losing a pet, you are not replacing this pet. You are keeping and treasuring memories, while filling the empty space in your family. Some people need to get another pet right away, some aren't ready for years.